A Worked Portfolio Optimization With Guardrails

From Segment Prices to a Portfolio Decision

A portfolio pricing decision can look attractive in each segment viewed alone and still fail an important portfolio-level requirement. Prices chosen to maximize contribution may, for example, reduce expected renewals below a required retention floor. A complete analysis therefore needs both an objective and a feasibility check: specify what the portfolio is trying to maximize, then determine whether the selected prices satisfy every guardrail.

This tutorial uses hypothetical portfolios with supplied linear demand curves. The curves are inputs to the pricing problem; the task is not to estimate customer response. The method is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For segment $j$, let $p_j$ be the premium per retained policy, $q_j(p_j)$ the expected number of retained policies at that premium, and $c_j$ the variable cost per retained policy. Expected contribution is $q_j(p_j)(p_j-c_j)$. A portfolio contribution objective is the sum of segment contributions when segment contributions are additive and there are no additional cross-segment effects or costs.

For a linear demand curve $q_j(p_j)=a_j-b_jp_j$, with $b_j>0$, the segment contribution is a concave quadratic:

\Pi_j(p_j)=(a_j-b_jp_j)(p_j-c_j) =-b_jp_j^2+(a_j+b_jc_j)p_j-a_jc_j.

Here, $c_j$ is assumed constant over the relevant price range, and the stated demand curve is assumed valid throughout the permitted price interval. The model treats $q_j$ as an expected count, so it need not be an integer. Fixed expenses, taxes, capital costs, and any effects not included in $c_j$ are excluded unless the problem explicitly supplies them.

Build the Objective and Guardrails

A price-change limit relative to an indicated premium becomes a lower and upper bound on the decision price. If the indicated premium is $r_j$ and the allowed dollar change is at most $h_j$, then $r_j-h_j\le p_j\le r_j+h_j$, subject to any separately stated absolute price limits. A retention floor is written using total expected retained policies, such as $\sum_j q_j(p_j)\ge R_{\min}$. These constraints should be written in the same price and demand units as the supplied data.

The unconstrained segment optimum is useful as a benchmark, not as an automatic portfolio answer. Differentiating the segment contribution gives

\Pi'_j(p_j)=a_j+b_jc_j-2b_jp_j, \qquad p_j^*=\frac{a_j+b_jc_j}{2b_j}.

The formula applies when the demand curve is linear, the marginal cost $c_j$ is constant, and the solution lies in the relevant price range. With price bounds, the best unconstrained price is clipped to the permitted interval. With a portfolio guardrail, even those bounded segment choices may not be jointly feasible.

Why the contribution formula holds: At a quoted price $p_j$, each retained policy contributes premium $p_j$ less variable cost $c_j$. The expected count of such policies is $q_j(p_j)$. By linearity of expectation, expected total contribution is expected count multiplied by contribution per policy, giving $q_j(p_j)(p_j-c_j)$. Adding segment expectations gives the portfolio objective when the segment contributions add.

A Reliable Solution Sequence

A useful order is to write the expected contribution and every constraint before calculating an optimum. Then check whether the constraints can be satisfied at all. For a retention floor and price intervals, the largest possible modeled retention occurs at the lowest permitted prices when each demand curve decreases with price. If that maximum is below the floor, the problem is infeasible; no optimization method can produce compliant prices.

Write the objective.
For each segment, multiply expected retained policies by contribution per policy, then sum across the portfolio.

Translate the guardrails.
Express price limits as bounds and retention requirements as inequalities in the decision prices.

Check feasibility.
Test whether the constraints can be met within the allowed price region, especially whether the retention floor is attainable.

Optimize and verify.
Find the best feasible prices, identify binding constraints, and recalculate contribution and retention at those prices.

Worked Portfolio Examples

Worked Example: A Binding Portfolio Retention Floor

A hypothetical insurer has two renewal segments. Segment A has expected retention $q_A(p_A)=120-2p_A$ and variable cost of $30 per retained policy. Segment B has $q_B(p_B)=100-p_B$ and variable cost of $20. Permitted prices are $35 to $45 in A and $25 to $40 in B. The portfolio must retain at least 100 expected policies. Find the contribution-maximizing feasible prices.

Method and objective. Expected portfolio contribution is

\Pi(p_A,p_B)=(120-2p_A)(p_A-30)+(100-p_B)(p_B-20).

Expanding each term gives $\Pi_A=-2p_A^2+180p_A-3600$ and $\Pi_B=-p_B^2+120p_B-2000$. The retention constraint is $220-2p_A-p_B\ge100$, or $2p_A+p_B\le120$, in addition to the two price intervals.

Check the independent benchmark. The unconstrained optimum in A is $(120+2(30))/(2(2))=45$, which is within A's interval. In B it is $(100+20)/(2(1))=60$, above B's maximum price, so the best price in B without the retention floor is $40. At these bounded prices, $q_A(45)=30$ and $q_B(40)=60$, totaling 90. That fails the minimum of 100; the independent segment choices are infeasible.

Find and verify the constrained prices. At $p_B=40$, B retains 60 policies, so A must retain at least 40. Since $120-2p_A\ge40$, A's price must be at most $40. A's contribution is increasing up to its unconstrained optimum of $45, so its best price under this requirement is $40. This gives $q_A=40$, $q_B=60$, and exactly 100 retained policies.

To check that allowing B's price to fall cannot improve the answer, at $(40,40)$ the contribution derivatives are $\Pi'_A=180-4(40)=20$ and $\Pi'_B=120-2(40)=40$. Along the active retention boundary, increasing $p_A$ by one dollar requires reducing $p_B$ by two dollars. The corresponding change in contribution per dollar of $p_A$ is $20-2(40)=-60$, so moving in that direction lowers contribution. Moving toward lower $p_A$ and higher $p_B$ is blocked by B's $40 price ceiling. The objective is concave and the constraints are linear, so this feasible boundary solution is the global maximum.

Result and trap avoided. The selected prices are $p_A=$$40 and $p_B=$$40. Contribution is $40(40-30)+60(40-20)=400+1{,}200=$$1,600. The common mistake is to report the independent choices of $45 and $40 without checking retention; those choices produce only 90 expected renewals and violate the guardrail.

Worked Example: Rate-Change Bounds with a Slack Retention Floor

In a hypothetical portfolio, segment C has demand $q_C(p_C)=150-3p_C$, variable cost $25, and an indicated premium of $36 with a permitted change of plus or minus $6. Segment D has $q_D(p_D)=90-1.5p_D$, variable cost $30, and an indicated premium of $36 with a permitted change of plus or minus $4. The portfolio must retain at least 60 expected policies. Maximize contribution.

Translate the limits and form the objective. The permitted intervals are $30\le p_C\le42$ and $32\le p_D\le40$. The objective is

\Pi=(150-3p_C)(p_C-25)+(90-1.5p_D)(p_D-30).

For C, differentiating gives $\Pi'_C=225-6p_C$, so the unconstrained optimum is $p_C=37.5$, inside its permitted interval. At this price, $q_C=150-3(37.5)=37.5$, and contribution is $37.5(37.5-25)=37.5(12.5)=$$468.75.

For D, $\Pi'_D=135-3p_D$, giving an unconstrained optimum of $p_D=45$, above the $40 ceiling. Contribution is increasing throughout the permitted interval, so the best permitted price is $40. It gives $q_D=90-1.5(40)=30$ and contribution $30(40-30)=$$300.

Check the portfolio guardrail. The selected prices retain $37.5+30=67.5$ expected policies, which exceeds the floor of 60. The retention constraint is therefore slack and does not require a price adjustment. Total contribution is $468.75 + $300 = $768.75. Prices C and D are, respectively, $1.50 and $4.00 above their indicated premiums, within the stated dollar-change limits.

Result and trap avoided. The selected prices are $p_C=$$37.50 and $p_D=$$40.00, with 67.5 expected retained policies and $768.75 contribution. Do not force a retention constraint to bind when it is already satisfied; doing so would change the problem and could lower the objective unnecessarily.

Worked Example: Feasibility Check and a Binding Segment Ceiling

A hypothetical insurer has segment A demand $q_A(p_A)=100-2p_A$, variable cost $20, and price bounds of $25 to $35. Segment B has $q_B(p_B)=80-p_B$, variable cost $10, and bounds of $20 to $35. The retention floor is 90 expected policies. Determine the optimal feasible prices and contribution.

Check feasibility first. Demand decreases as price increases, so maximum retention occurs at the lowest allowed prices: $q_A(25)=100-50=50$ and $q_B(20)=80-20=60$. Maximum total retention is 110, so the floor of 90 is attainable. At the opposite price corner, the total is $q_A(35)+q_B(35)=30+45=75$; that corner fails the floor.

Construct and assess the objective. The contribution is

\Pi=(100-2p_A)(p_A-20)+(80-p_B)(p_B-10).

The retention constraint $180-2p_A-p_B\ge90$ is equivalent to $2p_A+p_B\le90$. Without that constraint, A's unconstrained optimum is $(100+2(20))/4=35$; B's is $(80+10)/2=45$, limited to $35 by its ceiling. At those prices, retention is $30+45=75$, so the retention floor binds in the constrained problem.

At B's price ceiling of $35, the retention requirement becomes $2p_A+35\le90$, or $p_A\le27.5$. A's contribution increases up to $35, so the best price allowed by this requirement is $p_A=27.5$. It retains $100-2(27.5)=45$ policies; B at $35 retains $80-35=45$. Total retention is exactly 90.

Verify the segment ceiling is justified. At these prices the contribution derivatives are $\Pi'_A=140-4(27.5)=30$ and $\Pi'_B=90-2(35)=20$. The retention constraint's marginal cost in the A-price direction is twice its marginal cost in the B-price direction. Setting the A marginal contribution equal to twice the shadow value gives a shadow value of $30/2=15$. B's marginal contribution, $20, exceeds that shadow value; absent B's ceiling it would be worthwhile to raise $p_B$, but the $35 ceiling prevents doing so. The concave objective and linear constraints confirm a global constrained optimum at this boundary and ceiling.

Result and trap avoided. The selected prices are $p_A=$$27.50 and $p_B=$$35.00. Contribution is $45(27.50-20)+45(35-10)=337.50+1{,}125=$$1,462.50. The maximum-retention check established feasibility, while the 75-policy result at the independent bounded prices showed why simply optimizing each segment and stopping there would be wrong.

Interpretation and Practical Checks

The examples illustrate three distinct outcomes: a portfolio floor can force a price away from its independent optimum; a guardrail can be present but slack; and a binding retention requirement can interact with a segment price ceiling. These are different conditions, so report which constraints bind rather than describing every limit as an active restriction.

Before presenting a solution, verify the chosen prices against each price interval, calculate retention from the demand curves at those prices, and evaluate contribution from the original unexpanded formula. Also check the units: a retention floor of 100 expected policies is not a floor of 100 percent, and a dollar price-change cap is not a percentage cap. Linear demand may predict unrealistic or negative counts outside its applicable range; do not extrapolate beyond the stated model and price bounds.

Key takeaway: A portfolio solution is a price vector that maximizes the stated objective while satisfying every guardrail. Independent segment optima are only a starting benchmark: feasibility, binding constraints, and the final objective value must all be checked.

Check Your Understanding

For each hypothetical portfolio, write or check the objective, apply the stated bounds, and verify the guardrail. Answers give the requested final results.

Segment A has $q_A=100-2p_A$, cost $20, and bounds $25 to $40. Segment B has $q_B=80-p_B$, cost $10, and bounds $20 to $35. There is no portfolio retention constraint. Find the contribution-maximizing prices and total expected contribution. (Answer: $p_A=$$35 and $p_B=$$35; total contribution = $1,575.)
Segment A has $q_A=120-2p_A$, cost $30, and bounds $35 to $50. Segment B has $q_B=90-p_B$, cost $20, and bounds $25 to $40. Total retention must be at least 100. Find the selected prices and contribution. (Answer: $p_A=$$35 and $p_B=$$40; total contribution = $1,250, with 100 expected retained policies.)
Segment A has $q_A=200-4p_A$, cost $30, and bounds $30 to $45. Segment B has $q_B=120-2p_B$, cost $25, and bounds $25 to $45. The retention floor is 70. Find the selected prices, retention, and contribution, and state whether the floor binds. (Answer: $p_A=$$40 and $p_B=$$42.50; retention = 75 expected policies, so the floor is slack; contribution = $1,012.50.)
Segment A has $q_A=70-p_A$ with price bounds $30 to $40. Segment B has $q_B=90-2p_B$ with bounds $25 to $35. Before optimizing, assess whether a floor of 85 expected retained policies is feasible. (Answer: infeasible; maximum retention within the price bounds is $q_A(30)+q_B(25)=40+40=80$.)

What You'll Learn